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Achim Blumensath, Bruno Courcelle
To cite this version:
Achim Blumensath, Bruno Courcelle. On the monadic second-order transduction hierarchy. Logical Methods in Computer Science, Logical Methods in Computer Science Association, 2010, 6 (2), pp.1-28. �hal-00287223v3�
ACHIM BLUMENSATH AND BRUNO COURCELLE* TU Darmstadt, Germany
e-mail address: blumensath@mathematik.tu-darmstadt.de
Institut Universitaire de France and Bordeaux University, LaBRI, France e-mail address: courcell@labri.fr
Abstract. We compare classes of finite relational structures via monadic second-order transductions. More precisely, we study the preorder
C ⊑ K : iff C ⊆ τ(K) for some transduction τ .
If we only consider classes of incidence structures we can completely describe the resulting hierarchy. It is linear of order type ω + 3. Each level can be characterised in terms of a suitable variant of tree-width. Canonical representatives of the various levels are: the class of all trees of height n, for each n ∈N, of all paths, of all trees, and of all grids.
1. Introduction
Monadic second-order logic (MSO) is one of the most expressive logics for which the theories of many interesting classes of structures are still decidable. In particular, the infinite binary tree and many linear orders have a decidable MSO-theory [Rab69, She75] and the same holds for many classes of (finite or infinite) structures with bounded tree-width [BCL07, RS86]. Furthermore, for every fixedMSO-sentence ϕ and every class C of finite structures with bounded tree-width, there is a linear-time algorithm that checks whether a given structure from C satisfies ϕ [Bod96, FG06]. Examples of monadic second-order expressible graph properties are k-colourability, various types of connectivity, and planarity (via Kuratowski’s well-known characterisation by forbidden configurations).
A variant of monadic second-order logic called guarded second-order logic (GSO) allows quantification not only over sets of elements but also over sets of edges (i.e., tuples from the relations) [GHO02]. The above mentioned linear-time algorithms can be adapted to this logic. There are tight links between guarded second-order logic and tree-width: every class of (finite or infinite) relational structures with a decidableGSO-theory has bounded tree-width. This gives a sort of converse to the above mentioned decidability results [See91, Cou95]. The
1998 ACM Subject Classification: G.2.2 Graph Theory—Hypergraphs, F.4.1 Mathematical Logic—Model Theory.
Key words and phrases: Monadic Second-Order Logic, Guarded Second-Order Logic, Transductions, Hypergraphs.
* Supported by the GRAAL project of the ‘Agence Nationale pour la Recherce’.
LOGICAL METHODS
IN COMPUTER SCIENCE DOI:10.2168/LMCS-???
c
Achim Blumensath and Bruno Courcelle* Creative Commons
proof of this result uses a deep theorem of graph minor theory by Robertson and Seymour: a set of graphs has bounded tree-width if and only if it excludes some planar graph as a minor [RS86].
To compare the MSO-theories or GSO-theories of two classes of structures we can use monadic second-order transductions, a certain kind of interpretations suitable both, for monadic second-order logic and, using a detour via incidence structures, also for guarded second-order logic [BCL07, Cou91, Cou95, Cou03].
In the present article we classify classes of finite structures according to their ‘combi-natorial complexity’. (Note that we do not consider decidability issues.) We will consider two ways to measure the complexity of such classes. On the one hand, we can use their tree-width and its variants. On the other hand, we can compare them via transductions. As it turns out, these two approaches are equivalent and they give rise to the same hierarchy. This indicates the robustness of our definitions and their intrinsic interest. Other possible hierarchies, based on different logics, will be briefly mentioned in Section 9.
Let us give more details. An MSO-transduction is a transformation of relational struc-tures specified by monadic second-order formulae. As graphs can be represented by rela-tional structures, we can use MSO-transductions as transformations between graphs. An
MSO-transduction is a generalisation of the following kind of operations (see Definition 3.4): (i) the definition of a relational structure “inside” another one (in model theory this is called an interpretation);
(ii) the replacement of a structure A by the union of a fixed number of disjoint copies of A, augmented with appropriate relations between the copies;
(iii) the expansion of a given structure A by a fixed number of unary predicates, called parameters. Usually, these predicates are arbitrary subsets of the domain, but we also may have a formula imposing restrictions on them.
Because of the possibility to use parameters, a transduction τ is a many-valued map in general. (We may also think of it as non-deterministic.) Each relational structure A has several images τ (A, P1, . . . , Pn) depending on the choice of the parameters P1, . . . , Pn⊆ A.
If B = τ (A, P1, . . . , Pn) we can consider the tuple P1, . . . , Pn as an encoding of B in A. The
transduction τ is the corresponding decoding function.
Each transduction τ extends in a canonical way to a transformation between classes of structures. If C and K are classes of relational structures with C ⊆ τ (K), we can think of τ as a way of encoding the structures in C by elements of K. For instance, every finite graph can be encoded in a sufficiently large finite square grid (by a fixed transduction τ ). Every finite tree of height at most n (for fixed n) can be encoded in a sufficiently long finite path. But it is not the case that all finite trees can be encoded by paths (by a single transduction). The purpose of this article is to classify classes of finite relational structures according to their encoding powers. We will compare classes C and K of structures by the following preorder:
C ⊑ K : iff C ⊆ τ (K) for some MSO-transduction τ .
We attack the problem of determining the structure of this preorder. Since, at the moment, a complete description of this hierarchy seems to be out of reach, we concentrate on a variant where we replace monadic second-order logic by guarded second-order logic. In this case, the corresponding hierarchy can be described completely. To obtain a corresponding notion of transduction we cannot simply change the definition of an MSO-transduction to use GSO -formulae since the resulting notion of transduction would not yield a reduction between
GSO-theories, and it even would not be closed under composition. Instead, we will take a detour by combining ordinary MSO-transductions with a well-known translation between
GSOand MSO.
This translation is based on incidence structures. Let us first describe this notion for undirected graphs where it is very natural. There are two canonical ways to encode a graph G by a relational structure. We can use its adjacency representation which is a structure hV, edgi where the domain V consists of all vertices of G and edg is a binary relation containing all pairs of adjacent vertices. But we also can use the incidence representation of G. This is the structure hV ∪ E, ini where the domain V ∪ E contains both, the vertices and the edges of G, and in is the incidence relation between vertices and edges. In a similar way, we can associate with every relational structure A its incidence structure Ain (see
Definition 2.1) where the domain also contains elements for all tuples in some relation of A. It is shown in [GHO02] that everyGSO-formula ϕ talking about some structure A can be translated into anMSO-formula talking about the incidence structure Ain, and vice versa.
Hence, we can use incidence structures to obtain an analogue ⊑inof the preorder ⊑ suitable
for guarded second-order logic. We set
C ⊑inK : iff Cin⊆ τ (Kin) for someMSO-transduction τ ,
where Cin:= { Ain| A ∈ C }. The main result of the present article is a complete
characterisa-tion of the resulting hierarchy for classes of finite structures. We show that the preorder ⊑in
is linear of order type ω + 3. It turns out that every class of finite structures is equivalent to one of the following classes, listed in increasing order of generality:
• trees of height at most n, for each n ∈N; • paths;
• arbitrary trees (equivalently, binary trees); • (square) grids.
Each of these levels can be characterised in terms of tree decompositions. Hence, we also obtain a corresponding hierarchy of complexity measures on structures that are compatible withMSO-transductions transforming incidence structures.
The upper levels of the hierarchy can be determined easily using techniques from graph minor theory developed by Robertson and Seymour, such as the notions of a minor and a tree decomposition. In particular, we employ two results characterising bounded tree-width and bounded path-width in terms of excluded minors [RS83, RS86].
For the lower levels, which consist of classes of bounded path-width, the characterisation is more complicated and requires new results relating tree decompositions and monadic second-order logic.
In Sections 2 and 3 we give basic definitions. Section 4 collects some known results from graph minor theory. We also introduce a new variant of tree-width and prove some of its basic properties. In Section 5 we expound the connections between tree-width and monadic second-order transductions. In Section 6 we introduce the transduction hierarchy and we state our main theorem. Its proof is contained in Sections 7 and 8. In the first one, we prove that the hierarchy is strict while, in the second one, we show that it covers every class. The final Section 9 contains some extension of our results to other logics and some open problems in this direction.
2. Preliminaries
Let us fix our notation. We set [n] := {0, . . . , n − 1} and we write P(X) for the power set of a set X. We denote tuples ¯a with a bar. The components of ¯a will be a0, . . . , an−1
where the length n will usually be implicit. We sometimes identify a tuple ¯a with the set of its components. For instance, we write c ∈ ¯a to express that c = ai, for some i.
In this article all graphs, trees, and relational structures are finite. We will not repeat this finiteness assumption. A relational structure A is of the form hA, RA
0, . . . , R A
m−1i with
domain A and relations RA
i . The signature of such a structure is the set Σ = {R0, . . . , Rm−1}
of relation symbols. In some proofs we will also use signatures with constant symbols denot-ing elements of the domain. We write ar(R) for the arity of a relation R. For a signature Σ, we denote by STR[Σ] the class of all Σ-structures. We write A ⊕ B for the disjoint union of the structures A and B.
We mainly consider incidence structures. These are representations of structures A where we have added new elements to the domain, one for each tuple in the relations of A. Definition 2.1. Let A = hA, RA
0, . . . , R A
m−1i be a structure and let r be the maximal arity
of a relation Ri. The incidence structure of A is the structure
Ain:= hA ·∪ E, PR0, . . . , PRm−1, in0, . . . , inr−1i ,
where we extend the domain A by
E := RA0 ∪ · · · ∪ RA m−1,
and the relations are
PRi := { ¯c ∈ E | ¯c ∈ R
A i } ,
ini := { (a, ¯c) ∈ A × E | |¯c| > i and a = ci} .
The class of all incidence structures is STRin[Σ] := { Ain| A ∈STR[Σ] }.
Remark 2.2. Note that incidence structures are binary (i.e., their relations have arity at most 2). Hence, they can be regarded as bipartite labelled directed graphs.
One important property of incidence structures is the fact that they are sparse, i.e., their relations contain few tuples.
Definition 2.3. Let k ∈N. A structure A = hA, ¯Ri is k-sparse1if, for every subset X ⊆ A and all relations Ri, we have
Ri∩ Xar(Ri)
≤ k · |X| . Lemma 2.4. Every incidence structure is 1-sparse.
Let us fix our notation regarding trees and graphs.
Definition 2.5. A directed graph is a pair hV, edgi where V is the set of vertices and edg ⊆ V × V is the edge relation. Thus, graphs are by definition simple (without parallel edges). An undirected graph is a graph where the edge relation edg is symmetric. When speaking of a graph we will always mean an undirected one.
We regard a coloured graph as a relational structure hV, E0, . . . , Ek, P0, . . . , Pmi with
binary relations Ei and unary relations Pi that encode the colours of, respectively, the edges
and the vertices. We allow graphs whose edges and vertices have several colours.
1
[Cou03] introduced two notions of sparsity for hypergraphs: k-sparse hypergraphs and uniformly k-sparse hypergraphs. What we call k-sparse is a slight modification of the uniform version of [Cou03].
Trees play a major role in this article. Intuitively, a tree is a directed graph T with a unique vertex r of indegree 0, called the root of T, such that every vertex is reachable from r by a unique directed path. The actual definition we will use is slightly more concrete. It is based on the usual encoding of the vertices of a tree by finite sequences describing the path from the root to the given vertex. In fact, we introduce two notions of a tree: order-trees and successor-trees. The latter use the usual edge relation, while the former are equipped with the tree-order instead.
Definition 2.6. Let D be a set.
(a) For an ordinal α, we denote by D<α the set of all sequences of elements of D of length less than α. The prefix relation on D<ω is defined by
x y : iff y = xz , for some z ∈ D<ω.
The infimum of x and y with respect to , i.e., their longest common prefix, is denoted by x ⊓ y.
(b) A finite prefix closed subset T ⊆ D<ωis called a tree domain. Following our intuition
that a vertex is represented by the path leading to it, we call the empty sequence hi the root of T and the maximal elements of T its leaves. The domain of the complete m-ary tree of height n is m<n. Hence, m<0 is the empty tree, m<1 the one consisting only of the root, and m<2 consists of a root and m leaves.
Given a tree domain T we can define the successor relation edg on T by setting hx, yi ∈ edg : iff y = xd for some d ∈ D .
In this case we call y a successor of x and x the predecessor of y. A structure of the form hT, edgi (and every structure isomorphic to it) is called a successor-tree.
Sometimes it is convenient to replace the successor relation edg by the tree order . Structures of the form hT, i are called order-trees. A coloured tree is the expansion of a (order- or successor-) tree by unary predicates ¯P . (We do not require these predicates to be pairwise disjoint. Hence, every vertex may have none, one, or several colours.) We write TREEmfor the class of all order-trees hT, , P0, . . . , Pm−1i with m colours. The set of leaves
of a tree T is denoted by Lf(T).
(c) Let T = hT, i be an order-tree. The level of an element v ∈ T is the number of vertices u ∈ T with u ≺ v. We denote it by |v|. The height of T is the least ordinal α greater than the level of every element of T . Hence, the empty tree has height 0 and the tree with a single vertex has height 1. The out-degree of T is the maximal number of successors of a vertex of T. For successor-trees we define these notions analogously.
(d) Let T be a tree and v a vertex of T. The subtree of T rooted at v is the subtree Tv
consisting of all vertices u with v u, i.e., all vertices below v.
Sometimes it is possible to reduce statements about relational structures to statements about graphs. One way to do so consists in replacing a structure by its Gaifman graph. Definition 2.7. The Gaifman graph of a structure A = hA, ¯Ri is the undirected graph
Gf(A) := hA, edgi , with the same domain A and with the edge relation
3. Monadic second-order logic and transductions
Monadic second-order logic (MSO) is the extension of first-order logic by set variables and quantifiers over such variables. An important variant of MSO is guarded second-order logic (GSO) where one can quantify not only over sets of elements but also over sets of tuples from the relations (see [GHO02] for details). Hence, guarded second-order logic over a given structure A is equivalent to monadic second-order logic over its incidence structure Ain.
Lemma 3.1 ([GHO02]).
(a) For every GSO-sentence ϕ, we can effectively construct anMSO-sentence ψ such that A|= ϕ iff Ain|= ψ , for all structures A .
(b) For every MSO-sentence ϕ, we can effectively construct aGSO-sentence ψ such that Ain|= ϕ iff A|= ψ , for all structures A .
Throughout the article we will consistently work with incidence structures, thereby avoiding the treatment of guarded second-order logic. In particular, all formulae are tacitly assumed to beMSO-formulae.
Besides MSO and GSO we also consider their counting extensions CMSO and CGSO. These add predicates of the form |X| ≡ k (mod m) to, respectively, MSOand GSO, where X is a set variable and k, m are numbers. All of our results for GSO go through also for
CGSO, i.e., forCMSO-transductions between incidence structures. In Section 9 we will give a partial characterisation of the hierarchy forCMSO-transductions of graphs (not of incidence graphs). In this case the availability of counting predicates does make a difference.
To state the composition theorem below it is of advantage to work with a variant of
MSOwithout first-order variables. This variant has atomic formulae of the form X ⊆ Y and R ¯Z, for set variables X, Y, Z0, Z1, . . . , where a formula of the form R ¯Z states that there are
elements ai ∈ Zi such that the tuple ¯a is in R. Note that every general monadic second-order
formula with first-order variables can be brought into this restricted form by replacing all first-order variables by set variables and adding the condition that these sets are singletons.
Whenever we speak ofMSOwe will have this version in mind. In particular, the following definition of the rank of a formula is based on this variant. When writing down concrete for-mulae, on the other hand, we will allow the use of first-order variables to improve readability. We regard every such formula as an abbreviation of a formula of the restricted form. Simi-larly, when we use structures with constants we actually regard each constant as a singleton set.
Definition 3.2. (a) The rank qr(ϕ) of a formula ϕ is the nesting depth of quantifiers in ϕ. Formulae of rank 0 are called quantifier-free.
(b) The monadic theory of rank m of a structure A is
MThm(A) := { ϕ ∈MSO| A |= ϕ, qr(ϕ) ≤ m } .
For a tuple ¯a of elements of A, we also consider the monadic theory MThm(A, ¯a) of the
expansion hA, ¯ai.
Remark 3.3. We use the term ‘rank’ instead of the more natural ‘quantifier rank’ since in Section 9 below we will consider CMSO where the notion of rank has to be adapted for our results to go through.
In order to compare the monadic theories of two classes of structures we employ MSO -transductions. To simplify the definition we introduce three simple operations and we obtain
MSO-transductions as compositions of these.
Definition 3.4. (a) Let k ≥ 2 be a natural number. The operation copykmaps a structure A to the expansion
copyk(A) := hA ⊕ · · · ⊕ A, ∼, P0, . . . , Pk−1i
of the disjoint union of k copies of A by the following relations. Denoting the copy of an element a ∈ A in the i-th component of A ⊕ · · · ⊕ A by the pair ha, ii, we define
Pi := { ha, ii | a ∈ A } and ha, ii ∼ hb, ji : iff a = b .
For k = 1, we set copy1(A) := A.
(b) For m ∈ N, we define the operation expm that maps a structure A to all possible
expansions by m unary predicates Q0, . . . , Qm−1 ⊆ A. Note that this operation is
many-valued and that exp0 is just the identity.
(c) A basicMSO-transduction is a partial operation τ on relational structures described by a list
χ, δ(x), ϕ0(¯x), . . . , ϕs−1(¯x)
of MSO-formulae called the definition scheme of τ . Given a structure A that satisfies the formula χ the operation τ produces the structure
τ (A) := hD, R0, . . . , Rs−1i
where
D := { a ∈ A | A |= δ(a) } and Ri := { ¯a ∈ Dar(Ri)| A |= ϕi(¯a) } .
If A 6|= χ then τ (A) remains undefined.
(d) A k-copying MSO-transduction τ is a (many-valued) operation on relational struc-tures of the form τ0◦ copyk◦ expm where τ0 is a basic MSO-transduction. When the value
of k does not matter, we will simply speak of a transduction.
Note that, due to expm, a structure can be mapped to several structures by a transduc-tion. Consequently, we consider τ (A) as the set of possible values (τ0◦ copyk)(A, ¯P ) where
¯
P ranges over all m-tuples of subsets of A. For a class C, we set
τ (C) :=[{ τ (A) | A ∈ C } .
Remark 3.5. (a) The expansion by m unary predicates corresponds, in the terminology of [Cou95, Cou03], to using m parameters. We will use this terminology, for instance, in the proof of Theorem 5.5.
(b) Note that every basic MSO-transduction is a 1-copying MSO-transduction without parameters.
Example 3.6. (a) Let Σ be a signature and let r be the maximal arity of a relation in Σ. The operation mapping an incidence structure Ain∈STRin[Σ] to the structure Gf(A)in is a
k-copyingMSO-transduction where k = r(r − 1)/2, for r ≥ 2, and k = 1, for r ≤ 1.
(b) For every fixed number n ∈N, we describe a transduction τ transforming a path P of length l into the class of all trees of height n with l + 1 vertices.
We can encode a tree T of height n with m vertices as a finite word w of length m over the alphabet [n] as follows. Let v0 <lex · · · <lexvm−1 be the enumeration of the vertices of T
in lexicographic order, and let libe the level of vi. We encode T by the word w := l0. . . lm−1.
A transduction can recover T from w as follows. Each position in w corresponds to a vertex. The predecessor of the i-th vertex v is the maximal vertex to the left of v whose label is less than li. Clearly, this predecessor relation is definable in monadic second-order logic.
The two most important properties ofMSO-transductions are summarised in the follow-ing lemmas.
Lemma 3.7. Let τ be a transduction. For every MSO-sentence ϕ, there exists an MSO -sentence ϕτ such that, for all structures A,
A|= ϕτ iff B|= ϕ for some B ∈ τ (A) .
Furthermore, if τ is quantifier-free, then the rank of ϕτ is no larger than that of ϕ.
Corollary 3.8. For every quantifier-free transduction τ and every m ∈ N, there exists a function fτ on monadic theories of rank m such that
MThm(τ (A)) = fτ MThm(A), for all structures A .
Lemma 3.9 ([Cou91]). For all transductions σ, τ there exists a transduction ̺ such that ̺ = σ ◦ τ .
As a further example note that we can use transductions to translate between order-trees and successor-trees.
Lemma 3.10. (a) There exists a transduction τ mapping an order-tree to the corresponding successor-tree.
(b) There exists a transduction σ mapping a successor-tree to the corresponding order-tree.
Similarly there are transductions translating between a structure and its incidence struc-ture.
Lemma 3.11. For every signature Σ, there exists a transduction τ such that τ (Ain) = A,
for all A ∈STR[Σ].
The converse statement is a much deeper result and requires the structure in question to be k-sparse for some fixed k.
Theorem 3.12 ([Cou03, Blu10]). For every signature Σ and all numbers k ∈N, there exists an MSO-transduction τ such that τ (A) = Ain, for all k-sparse structures A ∈STR[Σ].
We have seen in Lemma 3.7 that transductions relate the monadic theories of two structures. We also need techniques to relate the monadic theory of a structure to those of its substructures. The disjoint union operation can frequently be used for this purpose (for a proof of the following theorem see, e.g., Theorem 7.11 of [Lib04], or [Cou87]).
Theorem 3.13. Let Σ and Γ be relational signatures with constants. For every m ∈ N, there exists a (computable) binary operation ⊕m on monadic theories of rank m such that
MThm(A ⊕ B) = MThm(A) ⊕mMThm(B) ,
Below we will mainly make use of the following corollary.
Lemma 3.14. Let T be an order-tree and v ∈ T a vertex. Suppose that T′ is the order-tree obtained from T by replacing the subtree Tv by some tree S. Let ¯c be a tuple of vertices of T
with v ci, for all i. If ¯a are vertices of Tv and ¯b are vertices of S such that
MThm(Tv, ¯a) = MThm(S, ¯b)
then it follows that
MThm(T, ¯a¯c) = MThm(T′, ¯b¯c) .
Proof Let C be the tree obtained from T by replacing the subtree Tv by a single vertex w.
We define the following auxiliary predicates:
P := {w} , Q0:= { x ∈ C | x ≺ w } , Q1 := Tv, and Q′1 := S .
We construct a quantifier-free transduction τ such that
τ hC, P, Q0, ¯ci ⊕ hTv, Q1, ¯ai= hT, ¯a¯ci ,
τ hC, P, Q0, ¯ci ⊕ hS, Q′1, ¯bi
= hT′, ¯b¯ci .
If fτ is the function from Corollary 3.8 and ⊕m the operation from Theorem 3.13, it follows
that MThm(T′, ¯b¯c) = fτ MThm(C, P, Q0, ¯c) ⊕mMThm(S, Q′1, ¯b) = fτ MThm(C, P, Q0, ¯c) ⊕mMThm(Tv, Q1, ¯a) = MThm(T, ¯a¯c) , as desired.
Hence, it remains to define τ . Let {, P, Q0, ¯d} be the signature of hC, P, Q0, ¯ci, and
{, Q1, ¯e} the signature of hTv, Q1, ¯ai and hS, Q′1, ¯bi. For τ we can use the basic MSO
-transduction consisting of the following formulae: χ := true , δ(x) := ¬P x ,
ϕ(x, y) := x y ∨ (Q0x ∧ Q1y) .
4. Minors and tree decompositions
Some properties of the transduction hierarchy, which we will introduce in Section 6 below, can be deduced from results about graph minors.
Definition 4.1. (a) Let G = hV, edgi be an undirected graph and E ⊆ edg a set of edges. We denote by E∗ the reflexive and transitive closure of E. Note that E∗ is an equivalence relation. The graph G/E is obtained by contracting all edges in E. Formally, we have
G/E := hW, edg0i ,
where W := V /E∗is the set of equivalence classes and edg0 contains an edge between classes [x] and [y] if and only if [x] 6= [y] and there are representatives u ∈ [x] and v ∈ [y] with hu, vi ∈ edg.
(b) A minor of a graph G is a graph that can be obtained from G by first deleting some vertices and edges and then contracting some of the remaining edges. For a class C of graphs, we denote by Min(C) the class of all minors of graphs in C.
One central tool in graph minor theory is the notion of a tree decomposition and the related complexity measures called tree-width and path-width. These notions extend in a natural way to relational structures.
Definition 4.2. Let A = hA, ¯Ri be a structure.
(a) A tree decomposition of A is a family D = (Uv)v∈T of (possibly empty) subsets
Uv ⊆ A indexed by a rooted tree T such that
• for every element a ∈ A, the set { v ∈ T | a ∈ Uv} is nonempty and connected in T ;
• for every tuple ¯c ∈ Ri, there is some index v ∈ T with ¯c ⊆ Uv.
We call the sets Uv the components of the decomposition, and T is its underlying tree.
The height of a tree decomposition D = (Uv)v∈T is the height of T , while its width is
the number
wd(D) := sup
v∈T
(|Uv| − 1) .
(b) The tree-width twd(A) of A is the minimal width of a tree decomposition of A. (c) The path-width pwd(A) of A is the minimal width of a tree decomposition of A where the underlying tree is a path.
(d) The n-depth tree-width twdn(A) of A is the minimal width of a tree decomposition
of A whose underlying tree has height at most n.
(e) For a class C of structures, we define twd(C) as the supremum of twd(A), for A ∈ C, and similarly for pwd(C) and twdn(C).
Remark 4.3. (a) The n-depth tree-width of a graph G is related to its tree-depth td(G) as introduced by Nešetřil and Ossona de Mendez [NdM06a, NdM06b]. The tree-depth of a graph G is the least number n such that some orientation of G is a subgraph of some order-tree of height n. For a graph G, it follows that
• td(G) ≤ n implies twdn(G) < n;
• twdn(G) < k implies td(G) ≤ nk.
These facts are easy to establish. We will not need them in the following.
(b) There are some simple relations between n-depth width, path-width, and tree-width. For every graph G, we have
twd(G) ≤ twdn+1(G) ≤ twdn(G) , for every n ∈N ,
and twd(G) = twdn(G) , for all sufficiently large n ∈N .
Furthermore,
pwd(G) < n(twdn(G) + 1) , for every n ∈N .
(Let us sketch the proof of the last inequality: let (Uv)v∈T be a tree decomposition of G of
height n and width twdn(G). As the components of a path decomposition of G we take all
sets of the form Uv0 ∪ · · · ∪ Uvk, where v0. . . vk is a path from the root v0 to some leaf vk
of T .)
The next lemma shows that most questions regarding tree decompositions of a structure can be reduced to the corresponding questions about its Gaifman graph. For many of the following results it is therefore sufficient to consider graphs.
Lemma 4.4. Let A be a structure. A family (Uv)v∈T is a tree decomposition of A if and
Proof (⇒) is immediate. (⇐) follows from the fact that every tree decomposition of a clique has one component Uv containing the whole clique. This implies that, for every clique C in
Gf(A) induced by some tuple ¯c ∈ Ri, there is some vertex v ∈ T with C ⊆ Uv. Hence, every
tuple ¯c ∈ Ri is contained in some component Uv.
In order to separate the higher classes of the hierarchy, we shall employ two deep results of Robertson and Seymour about excluded minors.
Theorem 4.5 (Excluded Tree Theorem [RS83]). For each tree T, there exists a number k ∈N such that
T∈ Min(G) implies/ pwd(G) < k , for every graph G .
Theorem 4.6 (Excluded Grid Theorem [RS86]). For each planar graph E, there exists a number k ∈N such that
E∈ Min(G) implies/ twd(G) < k , for every graph G .
Corollary 4.7. (a) A class of graphs has bounded path-width if and only if it excludes some tree as a minor.
(b) A class of graphs has bounded tree-width if and only if it excludes some planar graph as a minor.
We also need a variant of these theorems for n-depth tree-width. The next lemma contains the main technical argument.
Lemma 4.8. Suppose that G is a graph that does not contain a path of length l. Then G has a tree decomposition of height at most l and width at most l − 1.
Proof Let hT, i be a depth-first spanning order-tree of G, i.e., a spanning tree such that, for every edge (u, v) of G we have u v or v u (for details see, e.g., [Die06] where such spanning trees are called normal ). We define a tree decomposition (Uv)v∈T of G by setting
Uv := { u ∈ T | u v } .
Since T is depth-first, it follows that every edge (u, v) of G is contained in some compo-nent Uw where w is the maximum of u and v.
The height of the tree T can be at most l since G contains no path of length l. Further-more, we have |Uv| = |v| + 1 ≤ l. Hence, the width of the tree decomposition is at most
l − 1.
Theorem 4.9 (Excluded Path Theorem). For each path P, there exist numbers n, k ∈ N such that
P∈ Min(G) implies/ twdn(G) < k , for every graph G .
Proof Suppose that P /∈ Min(G) and let l be the length of P. Then the preceding lemma implies that twdl(G) < l.
Corollary 4.10. (a) A class of graphs has bounded n-depth tree-width, for some n, if and only if it excludes some path as a minor (equivalently, as a subgraph).
(b) A class of graphs has bounded tree-depth if and only if it excludes some path as a minor (equivalently, as a subgraph).
We can also compute a bound on the n-depth tree-width in terms of the (n + 1)-depth tree-width. It will be needed in the proof of Theorem 8.2 below.
We say that the tree hS, i can be embedded into a tree hT, i if there exists an order-preserving injective mapping hS, i → hT, i, i.e., if hS, i, regarded as relational structure, is isomorphic to an induced substructure of hT, i. For instance, we have an embedding
7→ 0 1 2 3 4 5 0 3 1 2 4 5
as indicated by the labels. If S can be embedded in T then S is isomorphic to a minor of T , when we consider S and T as graphs.
Definition 4.11. Let D = (Uv)v∈T be a tree decomposition and let F be a set of edges of
(the successor-tree corresponding to) T . The tree decomposition D/F obtained by contract-ing the edges in F is
D/F := (U[v]′ )[v]∈T /F ,
where U′ [v]:=
S
u∈[v]Uu.
Lemma 4.12. Let G be a graph and let D := (Uv)v∈T be a tree decomposition of G of
width k and height at most n + 1. If m ∈N is some number such that the tree m<n+1 cannot
be embedded into T , then twdn(G) < m(k + 1).
Proof We construct a tree decomposition D′of height at most n and width at most m(k+1)−1 as follows. Let P ⊆ T be the minimal (w.r.t. ⊆) set of vertices that contains
• every leaf of T at level n and
• every vertex that has at least m successors in P .
Since m<n+1 cannot be embedded into T it follows that P does not contain the root of T . Let F be the set of all edges of T linking a vertex in T r P to a vertex in P . By definition of P it follows that (i) every vertex of T has less than m F -successors; (ii) every path of T from the root to some leaf on level n contains at least one edge from F ; and (iii) no such path contains two consecutive edges from F .
The decomposition D′ := D/F obtained by contracting all edges in F has width at
most
k + 1 + (m − 1)(k + 1) − 1 < m(k + 1) . Furthermore, the height of the underlying tree is at most n.
Corollary 4.13. Let C be a class such that k := twdn+1(C) < ∞ and let m ∈ N. If every
structure A ∈ C has a tree decomposition (Uv)v∈T of width k and height at most n + 1 such
that the tree m<n+1 cannot be embedded into T , then twdn(C) < m(k + 1) < ∞.
We conclude this section with a lemma that will be useful when constructing transduc-tions τn that transform a structure into their tree decompositions of height n. Our
construc-tion works for all tree decomposiconstruc-tions that are strict in the following sense. Definition 4.14. Let (Uv)v∈T be a tree decomposition of a structure A.
(a) We define a function µ : A → T by
µ(a) := min { v ∈ T | a ∈ Uv} .
Note that µ(a) is well-defined since, by the definition of a tree decomposition, there is at least one v ∈ T such that a ∈ Uv.
(b) For v ∈ T , we set U⇑v := [ uv Uur [ u≺v Uu.
(c) The tree decomposition (Uv)v is strict if, for every v ∈ T ,
• Uv∩ µ(A) 6= ∅ (equivalently, Uvr Uu6= ∅, where u is the predecessor of v) and
• if v is not the root of T , then the subgraph of Gf(A) induced by the set U⇑v is
connected.
We conclude this section by a result implying that, for our purposes, it will be sufficient to consider only strict tree decompositions.
Lemma 4.15. Let G be a graph. For every tree decomposition (Uv)v∈T of G, there exists a
strict tree decomposition (Uv′)v∈T′ of G whose width and height are at most those of (Uv)v∈T.
Proof By induction on n ∈N, we will construct a sequence (Un
v)v∈Tn of tree decompositions
such that Un
⇑v is connected, for every v ∈ Tn with level 0 < |v| ≤ n. (Recall that |v| denotes
the level of v, and the root is the only vertex of level 0.) Furthermore, the restriction of Tnto
the set of vertices of level at most n will coincide with the corresponding restriction of Tn+1,
and we have Uvn+1= Uvn, for all v ∈ Tn+1 with level |v| ≤ n. It will follow that the sequence
has a limit (Uvω)v∈Tω where
Tω:=
[
n∈N
{ v ∈ Tn| |v| ≤ n } and Uvω:= Uv|v|.
We start the construction with T0 := T and Uv0 := Uv. Suppose that we have already
defined (Uvn)v∈Tn. For every vertex v ∈ Tn of level |v| = n + 1 we modify the tree
decompo-sition as follows. Let C0, . . . , Cm−1 be an enumeration of the connected components of U⇑vn .
We replace in Tn the subtree rooted at v by m copies S0, . . . , Sm−1 of the subtree, all
at-tached to the predecessor of v. For u ∈ Si we define Uun+1 := Uun∩ Ci. We can do these
modifications for all vertices of level n + 1 simultaneously. Let (Un+1
v )v∈Tn+1 be the resulting
tree decomposition.
The limit (Uvω)v∈Tω of this sequence satisfies the connectedness requirement of a strict
tree decomposition. To also satisfy the other condition we proceed as follows. Let F be the set of all edges (u, v) of Tω such that Uv∩ µ(V ) = ∅. (Note that this implies Uv ⊆ Uu.) We
construct the tree decomposition (U′
v)v∈T′ by contracting all edges in F . The details and
5. Tree decompositions and transductions
In this section we relate the material presented in the preceding one to monadic second-order transductions. Let us start by showing that there is a transduction computing the minors of a graph.
Lemma 5.1 ([Cou95]). There exists a transduction τ such that τ (Gin) = Min(G), for every
graph G.
Proof A minor H of G is obtained by deleting vertices, deleting edges, and contracting edges. Hence, we can encode H by four sets: the set of vertices we delete, the set of edges we delete, the set of edges we contract, and a set of vertices containing one representative of each contracted subgraph of G (these vertices serve as vertices of the resulting graph H). With the help of these parameters we can define H inside of Gin by MSO-formulae.
There is a close relationship between tree decompositions and transductions.
Lemma 5.2. For every signature Σ and every number k ∈ N, there exists a transduction τk:TREE0 →STRin[Σ] that maps an order-tree T to the class of all incidence structures Ain
such that the corresponding Σ-structure A has a tree decomposition of width at most k with underlying tree T .
Proof Suppose that A is a structure which has a tree decomposition (Uv)v∈T of width k. We
prove that A can be defined from a colouring of T where the number of colours depends only on Σ and k.
Let C0, . . . , Cm−1 be an enumeration of all Σ-structures whose domain is a subset of
[k + 1]. For each v ∈ T , let Uv be the substructure of A induced by Uv. It follows that, for
every v ∈ T , we can find some index λ(v) such that Uv ∼= Cλ(v). Let πv : Uv → Cλ(v) be the
corresponding isomorphism.
Furthermore, we associate with each edge (u, v) of T the binary relation R(u, v) := { (πu(a), πv(a)) | a ∈ Uu∩ Uv} ⊆ [k + 1] × [k + 1] .
We can recover A from T with the help of the vertex colouring λ and the edge colouring R. We form the disjoint union of all structures (Cλ(v))in, for v ∈ T , and we identify two elements
i ∈ Cλ(u) and j ∈ Cλ(v) if (u, v) is an edge of T such that (i, j) ∈ R(u, v). This can be performed by an n-copyingMSO-transduction where n is the maximal size of the structures (Ci)in, i < m.
We have just seen that we can map a class of trees to a class of structures with these trees as tree decompositions. Conversely, if we only consider strict tree decompositions, we can define a transduction mapping a class of structures to the corresponding class of trees. Recall the function µ : A → T from Definition 4.14. that assigns to an element a ∈ A the minimal index v ∈ T such that a ∈ Uv.
Proposition 5.3. For each number n ∈ N, there exists an MSO-formula ϕn(x, y; ¯Z) such
that, for every strict tree decomposition D = (Uv)v∈T of a graph G of height at most n, there
are sets L0, . . . , Ln−1 ⊆ V such that
G|= ϕn(a, b; ¯L) iff µ(a) ≤ µ(b) .
Proof Given D we use the sets
of all elements that first appear at level i of the tree. In particular, L0 = Uhi is the root
component of the tree decomposition. For k < n, let G≥k be the subgraph of G induced by
Lk∪ · · · ∪ Ln−1. For a ∈ Li and b ∈ Lj we define
a b iff i ≤ j and a, b belong to the same connected component of G≥i,
and a ∼ b iff a b and b a , or if a, b ∈ L0.
Clearly, the relation is MSO-definable with the help of the parameters ¯L. We claim that, for a, b /∈ L0, we have
a b iff µ(a) ≤ µ(b) .
(⇐) Suppose that µ(a) ≤ µ(b). Then b ∈ U⇑µ(a). Furthermore, U⇑µ(a) is connected since D is strict. Hence, U⇑µ(a) is a connected component of G≥i containing both a and b. Since
|µ(a)| ≤ |µ(b)| it follows that a b.
(⇒) Suppose that a b. Then there exists an undirected path π in G≥|µ(a)| connecting a and b. Since U⇑u∩U⇑v = ∅, for all u 6= v such that |u| = |v|, it follows that π is contained in
some U⇑vsuch that |v| = |µ(a)|. Since a is a vertex of π we must have v = µ(a). Furthermore,
b ∈ U⇑µ(a) since b is also a vertex of π. This implies that µ(a) ≤ µ(b).
Theorem 5.4. For each constant n ∈N, there exists a transduction τn mapping a graph G
to the class of all (underlying trees of ) strict tree decompositions of G of height at most n. Proof Let D = (Uv)v∈T be a strict tree decomposition of G, and let ϕn(x, y; ¯L) be the
formula from Proposition 5.3 with parameters L0, . . . , Ln−1⊆ V . We can define the tree T
underlying D as follows:
• Its root is any element of L0 = Uhi.
• For the other vertices of T , we choose one vertex in each ∼-class different from L0.
Note that ∼ is definable with the help of ϕn.
• The ordering of T is defined by ϕn.
Hence, we obtain a transduction with parameters ¯L that transforms a graph into a ‘candidate’ tree decomposition. Via a backwards translation we can write down a formula stating that the candidate given by the parameters ¯L corresponds to an actual strict tree decomposition. We omit the details which are standard for this type of construction.
In Lemma 5.2 we have seen how to obtain classes of bounded tree-width from classes of trees. Conversely, it is the case that every class obtained from a class of trees via a transduction has a bounded tree-width.
Theorem 5.5. For every transduction τ : TREEm → STRin[Σ], there exists a number
k ∈N such that, for each m-coloured tree T with image Ain ∈ τ (T), the structure A has a
tree decomposition of width at most k where the underlying tree is T.
Remark 5.6. (a) Courcelle and Engelfriet [CE95] have shown that an incidence struc-ture Ain obtained via a transduction τ from an m-coloured tree T has bounded tree-width.
Theorem 5.5 strengthens this result by proving that, if Ain is the image of a tree T, then we
can use the same tree T as the tree underlying a tree decomposition of the given width. (b) Lapoire has announced in [Lap98] a result somewhat related to Theorem 5.4. He claims that, for every k ∈ N, there exists a transduction that transforms a given graph G of tree-width at most k to a coloured tree (like in the proof of Lemma 5.2) that encodes some tree decomposition of G of width at most k. Our result is less ambitious in the sense
that we only consider tree decompositions of a fixed height. This enables us to give a precise description of which tree decompositions (the strict ones) our transduction returns. Note that one can show that, for k ≥ 2, there is no such transduction that would return all tree decompositions of G of width at most k.
We split the proof into several lemmas. As a technical tool we introduce a second kind of hierarchical decompositions of structures and a corresponding notion of width. To simplify the definition we will only consider incidence structures.
Definition 5.7. Let Ain= hA ∪ E, ¯P , in0, . . . i be an incidence structure.
(a) A partition refinement of Ain is a family Π = (Wv, ≈v)v∈T of pairs consisting of a
subset Wv ⊆ A ∪ E and an equivalence relation ≈v on Wv with the following properties:
• The index set T is a tree.
• For the root hi, we have Whi= A ∪ E
• For every internal vertex (i.e., non-leaf) u ∈ T with successors v0, . . . , vn−1, the sets
Wv0, . . . , Wvn−1 form a partition of Wu.
• |Wu| = 1, for every leaf u ∈ T .
• x ≈v y and u v implies x ≈uy.
• If u is an internal vertex of T , v, w successors of u, not necessarily distinct, and x ∈ Wv, y ∈ Ww elements, then x ≈u y implies either
x, y ∈ A and, for every e ∈ Er (Wv∪ Ww) and every i ,
(x, e) ∈ ini ⇔ (y, e) ∈ ini
or x, y ∈ E and, for every a ∈ Ar (Wv∪ Ww) and every i ,
(a, x) ∈ ini ⇔ (a, y) ∈ ini.
Note that it follows that, for every element x ∈ A ∪ E, there is some leaf u ∈ T such that Wu = {x}.
(b) The width of a partition refinement Π = (Wv, ≈v)v∈T is the maximum number of
equivalence classes realised in some component Wv:
wd(Π) := max
v∈T |Wv/≈v| .
The partition-width of the structure Ainis the minimal width of a partition refinement of Ain.
The notion of a partition refinement and of partition-width are adaptations of definitions from [Blu06, Blu03]. Up to a factor of 2, the partition-width of an incidence structure and its clique-width coincide.
Example 5.8. Let A = (A, R) be a structure with domain A = {a, b, c, d, e} and a ternary relation R = {(a, b, c) | {z } x , (a, b, d) | {z } y , (a, b, e) | {z } z } .
Its incidence structure is Ain = hA ∪ E, PR, in0, in1, in2i with E = {x, y, z}. We obtain a
partition refinement
{a | b | c, d, e | x, y, z}
{a | b} {x | c} {y | d} {z | e}
where we have indicated the partition into ≈v-classes by vertical bars. This partition
refine-ment has width 4.
Lemma 5.9. For every partition refinement Π = (Wv, ≈v)v∈T of an incidence structure
Ain = hA ∪ E, ¯P , in0, . . . , inr−1i, there exists a tree decomposition D = (Uv)v∈T of A with
the same underlying tree T such that
wd(D) < (r + 3) · wd(Π) .
Proof Let l : A ∪ E → Lf(T ) be the function assigning to every x ∈ A ∪ E the unique leaf l(x) of T such that Wl(x)= {x}. We claim that the desired tree decomposition (Uu)u∈T of A
is given by
Uu := Bu∪ Cu∪ Du
where
Bu := { v ∈ A | u l(v) and (v, e) ∈ ini for some i < r and e ∈ E with u l(e) } ,
Cu := { v ∈ A | u l(v) and (v, e) ∈ ini for some i < r and e ∈ E with u l(e) } ,
Du := { v ∈ A | (v, e) ∈ ini for some i < r and e ∈ E with l(v) ⊓ l(e) = u } .
Note that the connectedness condition holds since (v, e) ∈ ini implies that v belongs to
precisely those components Uu such that u lies on the path from l(v) to l(e).
It remains to prove that |Uu| ≤ (r + 3) · wd Π. If u = l(¯c), for some ¯c ∈ E, then Uu = Cu
consists of the components of ¯c. Hence, |Uu| = |¯c| ≤ r. Therefore, we may assume that
u /∈ l[E]. Let
[x]u := { y ∈ Wu| y ≈u x }
denote the ≈u-class of x. We prove the following bounds.
(1) |[x]u| = 1, for all x ∈ Bu.
(2) |[x]u∩ Uu| ≤ 2, for all x ∈ Du.
(3) |Cu| ≤ r · |Wu/≈u|.
Since Bu, Du⊆ Wu it then follows that |Uu| = |Bu∪ Cu∪ Du| ≤ (r + 3) · |Wu/≈u|.
(1) Let x ∈ Bu. There is some tuple e ∈ E and some index i such that (x, e) ∈ ini and
u l(e). We have (y, e) ∈ ini, for every y ∈ Wu such that y ≈u x. Since x is the only such
element it follows that [x]u= {x}.
(2) Let x ∈ Du. There is some tuple e ∈ E and some i such that (x, e) ∈ ini and
u = l(x) ⊓ l(e). Let v be the successor of u such that v l(e). We have (y, e) ∈ ini, for all
y ∈ Wur Wv such that y ≈u x. Hence, [x]ur Wv = {x}.
Suppose that there is some element y ∈ [x]u∩ Wv∩ Uu. By definition of Uuthere is some
tuple f ∈ E and some j such that (y, f ) ∈ inj and l(y) ⊓ l(f ) u. As above it follows that
[x]u∩ Wv = {y}. Consequently, we have |[x]u∩ Uu| ≤ 2.
(3) Let x ∈ Cu and consider some tuple e ∈ E such that (x, e) ∈ ini and u l(e). Set
Iu(e) := { z ∈ A | (z, e) ∈ ini for some i and u l(z) } .
For e, f ∈ E ∩ Wu, it follows that
e ≈uf implies Iu(e) = Iu(f ) .
Furthermore, we obviously have |Iu(e)| ≤ |e| ≤ r. It follows that Cu contains at most
Lemma 5.10. Let τ : TREEm → STRin[Σ] be a basic MSO-transduction such that, for
every m-coloured order-tree T with image Ain∈ τ (T), we have
A ∪ E = Lf(T) and A ∩ E = ∅ .
Then there exists a number n ∈N such that, for every order-tree T, we can find a partition refinement (Wv, ≈v)v∈T of τ (T) of width at most n.
Proof Let hχ, δ(x), (ϕPR(x))R, (ϕini(x, y))i<ri be the definition scheme of τ , and let h be the
maximal rank of these formulae.
Given T we define the desired partition refinement Π = (Wu, ≈u)u∈T by setting
Wu := { x ∈ Lf(T ) | u x } ,
and x ≈u y : iff MThh(Tv, x) = MThh(Tw, y) ,
where v, w are the successors of u with x ∈ Wv and y ∈ Ww.
(If u is a leaf of T then Wu = {x} and we take the equality relation for ≈u.) Note that the
index of ≈v is finite and that it only depends on h and not on the input tree T.
It remains to show that Π is actually a partition refinement. First, let us prove that x ≈v y and u v implies x ≈u y. It is sufficient to consider the case that u is the predecessor
of v. Then the general case follows by induction. Hence, suppose that v is a successor of u, that w, w′ are successors of v, and that x, y are leaves with w x and w′ y such that x ≈v y. Then we have
MThh(Tw, x) = MThh(Tw′, y) ,
which, by Lemma 3.14, implies that
MThh(Tv, x) = MThh(Tv, y) .
Consequently, we have x ≈u y.
We also have to show that the incidence relation is invariant under ≈u. Let v, w be
successors of u and suppose that x, y are leaves with v x and w y such that x ≈uy. We
distinguish two cases.
Suppose that x, y ∈ A and let e ∈ Er (Wv∪ Ww) be an edge. Since
MThh(Tv, x) = MThh(Tw, y) ,
it follows that
T|= ϕini(x, e) iff T|= ϕini(y, e) .
Hence, (x, e) ∈ ini iff (y, e) ∈ ini.
Now, suppose that x, y ∈ E and let z ∈ Ar (Wv∪ Ww) be an element. Since
MThh(Tv, x) = MThh(Tw, y) ,
it follows that
T|= ϕini(z, x) iff T|= ϕini(z, y) .
Proof of Theorem 5.5 (1) First, suppose that τ : TREEm → STRin[Σ] is a basic MSO
-transduction such that, for every m-coloured order-tree T ∈ dom(τ ) with image Ain∈ τ (T),
we have
A ∪ E = Lf(T) and A ∩ E = ∅ .
It follows by Lemma 5.10 that there is a number w ∈N such that, for every tree T, we can find a partition refinement of Ain∈ τ (T) with underlying tree T whose width is at most w.
By Lemma 5.9 it follows that A has a tree decomposition (Uv)v∈T with underlying tree T
and whose width is less than k := w(r + 3). (2) If τ is a basic MSO-transduction such that
A ∪ E ⊆ Lf(T) and A ∩ E = ∅ , for all Ain∈ τ (T) with T ∈ dom(τ ) ,
then we can argue similarly. Let τ′ be the MSO-transduction mapping T to the structure
obtained from τ (T) by adding one isolated element for every leaf of T that does not corre-spond to an element of τ (T). Then τ′ is of the form considered in (1) and we obtain a tree
decomposition (Uv)v∈T of τ′(T). Deleting from every component Uv all elements not in τ (T)
we obtain the desired tree decomposition of τ (T).
(3) Suppose that τ is a non-copyingMSO-transduction as in (2) but with p parameters. We can regard τ as a basicMSO-transductionTREEm+p→STRin[Σ]. By (2) it follows that,
for every value of the parameters ¯P , the structure τ (T, ¯P ) has a tree decomposition of the required form.
(4) Finally, consider the general case. Suppose that τ is l-copying. Given T let T+ be
the tree obtained from T by adding l new successors to every vertex of T. Formally, suppose that T ⊆ D<ω, for some finite set D. W.l.o.g. we may assume that D ∩ [l] = ∅. We define
the domain T+⊆ (D ∪ [l])<ω of T+ by
T+ := T ∪ (T × [l]) . Furthermore, we add new colour predicates
Si := T × {i} , for i ∈ [l] .
Note that every element of τ (T) is of the form hv, ii where i ∈ [l] and v ∈ T . Hence, each such element corresponds to a leaf vi ∈ T × [l] ⊆ T+. Using the parameters ¯S we can
construct a basic MSO-transduction τ+ : TREEm+l → STRin[Σ] satisfying the conditions
of (3) such that τ+(T+) = τ (T). By (3), we obtain a tree decomposition H+ = (Uv+)v∈T+
of τ+(T+) = τ (T). Let H = (U
v)v∈T be the tree decomposition obtained from H+ by
contracting every edge leading to a leaf in T+r T . Then we have wd(H) + 1 ≤ (l + 1)(wd(H+) + 1)
and wd(H+) ≤ w , for some w ∈N independent of T .
6. The transduction hierarchy
The focus of our investigation lies on the following preorder on classes of structures which compares their ‘encoding powers’ with respect toMSO-transductions. Our main result is a complete description of the hierarchy induced by this preorder. It will be given in Theorem 6.4.
(a) C ⊑ K if there exists a transduction τ such that C ⊆ τ (K). (b) C ⊏ K if C ⊑ K and K 6⊑ C.
(c) C ≡ K if C ⊑ K and K ⊑ C.
(d) C ⊳ K if C ⊏ K and there is no class D with C ⊏ D ⊏ K. (e) C ⊑inK if Cin⊑ Kin.
(f) The relations ⊏in, ≡in, and ⊳in are defined analogously to ⊏, ≡, ⊳ by replacing ⊑
everywhere by ⊑in.
The transduction hierarchy is the hierarchy of classes C ⊆STR induced by the relation ⊑in.
As the class of transductions is closed under composition, it follows that the relation ⊑in
is a preorder, i.e., it is reflexive and transitive. Lemma 6.2. ⊑in is a preorder on P(STR).
Definition 6.3. We consider the following subclasses of STR[{edg}]. (All trees below are considered to be successor-trees.)
(a) Tn:= { m<n| m ∈N } is the set of all complete m-ary trees of height n.
(b) Tbin is the class of all binary trees.
(c) Tω is the class of all trees.
(d) P is the class of all paths.
(e) G is the class of all rectangular grids.
The following description of the transduction hierarchy is the main result of the present paper.
Theorem 6.4. We have the following hierarchy:
∅ ⊳inT0⊳inT1 ⊳in. . . ⊳inTn⊳in· · · ⊏inP ⊳inTω ≡inTbin⊳inG
For every signature Σ, every class C ⊆ STR[Σ] is ≡in-equivalent to some class in this
hierarchy.
Remark 6.5. There is a lot of flexibility in the choice of representatives for the various levels. For instance, we could replace Tn by the class of all trees of height at most n, Tω by
Tbin or { 2<n| n ∈N }, and G by the class of square grids.
It is straightforward to show that the above classes form an increasing chain. The hard part is to prove that the chain is strictly increasing and that there are no further classes. Lemma 6.6. We have
∅ ⊑inT0 ⊑inT1⊑in· · · ⊑inTn⊑in· · · ⊑inP ⊑inTω⊑inG .
Proof In the example before Lemma 3.7, we have constructed transductions τn such that
Tn ⊆ τn(P). Hence, Tn ⊑in P. The remaining assertions follow from the observation that,
by Lemma 5.1, C ⊆ Min(K) implies C ⊑inK.
Let us collect some easy properties of the hierarchy. Our first result states that G is a representative of the top level of the transduction hierarchy.
Lemma 6.7. STR[Σ] ⊑inG
Proof Recall that the m × n grid is the undirected graph G = hV, edgi with vertices V = [m] × [n] and edge relation
Before encoding arbitrary structures in such grids we describe a transduction mapping G to its directed variant hV, E0, E1i where
E0 := { (hi, ki, hi + 1, ki) | i < m − 1, k < n } ,
and E1 := { (hi, ki, hi, k + 1i) | i < m, k < n − 1 } .
This can be done with the help of the parameters P0, P1, P2, Q0, Q1, Q2⊆ V where
Pm:= { hi, ki | i ≡ m (mod 3) } ,
and Qm:= { hi, ki | k ≡ m (mod 3) } .
Then
E0 = { (u, v) ∈ edg | u ∈ Pi and v ∈ Pj for some i ≡ j − 1 (mod 3) } ,
and E1 = { (u, v) ∈ edg | u ∈ Qi and v ∈ Qj for some i ≡ j − 1 (mod 3) } .
It is easy to write down a formula checking that the parameters Pm and Qm are correctly
chosen (see, e.g., [Cou97]).
To show that STR[Σ] ⊑in G, suppose that A ∈ STR[Σ] is a structure with Ain =
hA ∪ E, (PR)R, in0, . . . , inr−1i. Fix enumerations a0, . . . , am−1of A and e0, . . . , en−1 of E. By
the above remarks, it is sufficient to encode Ain in the directed m × n grid. Consider the
following subsets of [m] × [n]:
A′ := [m] × {0} , PR′ := { h0, ki | ek ∈ PR} ,
E′ := {0} × [n] , Il′ := { hi, ki | (ai, ek) ∈ inl} .
Then Aincan be recovered from G by anMSO-transduction using these sets as parameters.
Lemma 6.8. Tω≡inTbin.
Proof For one direction, note that Tbin⊆ Tω implies Tbin⊑inTω. Conversely, each finite tree
can be obtained as minor of a binary tree. Therefore, we have Tω⊆ Min(Tbin) ⊑inTbin.
Lemma 6.9. We have C ≡inT1 if and only if C is finite and contains at least one nonempty
structure.
As indicated in the example before Lemma 3.7, there exists a transduction mapping an incidence structure Ain to the incidence structure Gf(A)in of the Gaifman graph of A.
Lemma 6.10. For every class C of structures, we have Gf(C) ⊑inC .
The next result is just a restatement of Lemma 5.1 in our current terminology. Lemma 6.11. For every class C of graphs we have Min(C) ≡inC .
7. Strictness of the hierarchy
In this section we prove that the hierarchy is strict. Using the results of Section 5 we can characterise each level of the hierarchy in terms of tree-width and its variants.
Theorem 7.1. Let C ⊆STR[Σ]. (a) C ⊑inP iff pwd(C) < ∞.
(b) C ⊑inTω iff twd(C) < ∞.
(c) C ⊑inTn iff twdn(C) < ∞.
Corollary 7.2. Let C be a class of Σ-structures. (a) pwd(C) = ∞ implies Tω⊑inC.
(b) twd(C) = ∞ implies G ⊑inC.
Proof (a) Suppose that pwd(C) = ∞. Then Theorem 4.5 implies that Tω ⊆ Min(Gf(C)).
Hence, the claim follows from Lemmas 6.10 and 6.11.
(b) Suppose that twd(C) = ∞. Then Theorem 4.6 implies that G ⊆ Min(Gf(C)). As in (a), the claim follows from Lemmas 6.10 and 6.11.
Corollary 7.3. Let C ⊆STR[Σ]. (a) C 6⊑inP implies Tω ⊑inC.
(b) C 6⊑inTω implies G ≡inC.
In particular, it follows that the upper part of the hierarchy is strict: Corollary 7.4. P ⊳inTω⊳inG
Proof Both assertions follow from Theorem 7.1 and Corollary 7.3.
For the first one, note that we have P ⊑inTωsince P ⊆ Min(Tω). Conversely, pwd(Tω) =
∞ implies, by Theorem 7.1 (a), that Tω 6⊑in P. Hence, P ⊏in Tω. Finally, if C ⊏in Tω then
C ⊑inP, by Corollary 7.3 (a). Consequently, we have P ⊳inTω.
Similarly, the fact that Tω ⊑in G follows from Lemma 6.7. Since twd(G) = ∞,
Theo-rem 7.1 (b) implies that Tω⊏inG. Finally, we obtain Tω ⊳inG by Corollary 7.3 (b). Let us turn to the lower part. We start with two technical lemmas.
Definition 7.5. Let T = hT, ≤i be an order-tree. Vertices v0, . . . , vm−1 of T are horizontally
related via a vertex w if all vi are at the same level of the tree and vi ⊓ vk = w, for all
0 ≤ i < k < m.
Lemma 7.6. Let T be a coloured order-tree of height n, and suppose that τ is a parameterless k-copying MSO-transduction of rank r such that τ (T) is a successor-tree of height at most n + 1. Consider vertices v0, . . . , vm−1 of T that are horizontally related via w and fix some
number l < k. Let xi be the successor of w with xi vi. If, for all i, j < m, we have
MThr+2n+1(Txi, vi) = MThr+2n+1(Txj, vj) ,
then at least m − 1 elements of the set {hv0, li, . . . , hvm−1, li} (these are elements of the
domain of τ (T)) are horizontally related in τ (T).
Proof Let ϕss′(x, y) be the formula defining the successor relation in τ (T) between vertices
of the form hx, si and hy, s′i. By assumption the rank of ϕ
ss′(x, y) is at most r.
First, note that a vertex hv, li is on level h in τ (T) if and only if there are indices s0, . . . , sh−1< k such that
T|= ψs0...sh−1(v)
where the formula
ψs0...sh−1(v) := ∃y0· · · ∃yh−1 h ^ i<h−1 ϕsisi+1(yi, yi+1) ∧ ϕsh−1l(yh−1, v) ∧ ¬∃z _ s<k ϕss0(z, y0) i
expresses that there exists a path of the form hy0, s0i, . . . , hyh−1, sh−1i, hv, li from the root
hy0, s0i of τ (T) to the vertex hv, li. By assumption on vi and Lemma 3.14, we have
MThr+2n+1(T, vi) = MThr+2n+1(T, vj) , for all i, j .
Since the rank of ψs0...sh−1 is h + r + 1 ≤ r + 2n + 1 it follows that
T|= ψs0...sh−1(vi) iff T|= ψs0...sh−1(vj) .
Hence, all vertices hv0, li, . . . , hvm−1, li are on the same level h in τ (T). We prove by induction
on h that
(∗) MThr+n+h+1(Txi, vi) = MThr+n+h+1(Txj, vj)
implies that all but at most one of hv0, li, . . . , hvm−1, li are horizontally related.
Let hui, sii be the predecessor of hvi, li in τ (T), that is,
T|= ϕsil(ui, vi) .
We distinguish two cases.
First suppose that u0⊓ v0 w in T. By (∗) and Lemma 3.14, we have
MThr+n+h+1(T, u0, v0) = MThr+n+h+1(T, u0, vi) ,
for all i such that u0⊓ vi w. Note that there can be at most one index i that does not
satisfy this condition since, if we had u0⊓ vi w and u0⊓ vj w, for i 6= j, then we would
have vi⊓ vj ≻ w and v0, . . . , vm−1 would not be horizontally related via w. It follows that
T|= ϕs0l(u0, v0) implies T|= ϕs0l(u0, vi) , for all i as above .
Hence, hu0, s0i is the common predecessor of all the hvi, li, except for possibly one of them.
(For an index i with u0⊓ vi w our composition argument does not work since in that case
(∗) does not imply that the theories of (T, u0, v0) and (T, u0, vi) coincide.)
It remains to consider the case that w ≺ u0⊓ v0. Setting
ηui :=
^
MThr+n+h−1+1(Txi, ui)
we have
Tx0 |= ∃z[|z| = |u0| ∧ ηu0(z) ∧ ϕs0l(z, v0)] .
Since the rank of this formula is r + n + h + 1 it follows that
Txi |= ∃z[|z| = |u0| ∧ ηu0(z) ∧ ϕs0l(z, vi)] , for all i < m .
Consequently, we have |ui| = |u0|, for all i, and u0, . . . , um−1 are horizontally related via w.
Since the vertices hu0, s0i, . . . , hum−1, s0i are on level h−1 in τ (T), we can apply the induction
hypothesis and it follows that all but at most one of then are horizontally related via some vertex w′. Therefore, the same holds for their successors hv
Definition 7.7. We denote by B(n, k, c) the number of functions m<n → P([c]) with m ≤ k. Intuitively, each such function corresponds to a vertex-colouring of the tree m<n with c colours.
Lemma 7.8. For n ≥ 1 and k ≥ 2, we have
2ckn−1 ≤ B(n, k, c) ≤ k22ckn−1. Proof For m ≥ 2, we have
mn−1≤ mn−1+ X i<n−1 mi = mn−1+m n−1− 1 m − 1 ≤ 2m n−1.
Since [m]<n =Pi<nmi = mn−1+Pi<n−1mi it follows that mn−1≤ [m]<n ≤ 2mn−1. Therefore, we can bound
B(n, k, c) = 2cn+ k X m=2 2c|[m]<n| from above by B(n, k, c) ≤ 2cn+ k X m=2 2c2mn−1 ≤ k22ckn−1
and from below by
B(n, k, c) ≥ 2cn+ k X m=2 2cmn−1 ≥ 2ckn−1. Theorem 7.9. Tn⊏inTn+1
Proof For a contradiction, suppose that there exists a transduction τ such that (Tn+1)in⊆
τ ((Tn)in). Let Tnord be the class of all order-trees corresponding to successor-trees in Tn, and
let Tn+1col := exp1(Tn+1) be the class of all coloured successor-trees with one colour whose
underlying tree is in Tn+1. Since the successor-trees in Tn are 1-sparse we can construct an MSO-transduction σ0 such that (Tn)in⊆ σ0(Tn). Since Tnord ≡ Tn we can combine τ and σ0
to a transduction σ such that Tcol
n+1 ⊆ σ(Tnord). By Lemma 7.6, it follows that there is some
constant d such that every tree T ∈ σ(Tnord) with out-degree at most k is of the form σ(T′) where T′ ∈ Tord
n has out-degree at most dk. (The out-degree of an order-tree is the out-degree
of the corresponding successor-tree.) Suppose that σ uses c parameters. There are B(n, dk, c) ≤ dk22c(dk)n−1
colourings of trees in Tnord with out-degree at most dk. On the other hand, there are B(n + 1, k, 1) ≥ 2kn
trees in Tn+1col with out-degree at most k. For large k it follows that B(n, dk, c) ≤ dk22cdn−1kn−1 < 2kn = B(n + 1, k, 1) . Consequently, there is some tree in Tcol
n+1 that is not the image of a tree in Tnord. A
8. Completeness of the hierarchy
We have shown that the hierarchy presented in Theorem 6.4 is strict. To conclude the proof of the theorem it therefore remains to show that there are no additional classes. We have already seen in Corollary 7.4 that Tω and G are the only classes above P. Next we shall
prove that Tn⊳Tn+1.
Lemma 8.1. Let C be a class of structures. If, for every number m ∈N, there exists a struc-ture A ∈ C such that we can embed m<n+1 into every tree underlying a tree decomposition (Uv)v∈T of A of width k, then Tn+1⊑inC.
Proof By Lemma 4.15, it follows that, for every m ∈ N, there is a structure in C with a strict tree decomposition of width at most k and with an underlying tree T into which we can embed the tree m<n+1. According to Proposition 5.3 there is a transduction mapping C to the class of trees underlying these strict tree decompositions. Hence, there exists a class K ⊑inC of successor-trees containing, for every m ∈N, some tree into which we can embed
m<n+1. Hence, T
n+1⊆ Min(K) ⊑inK ⊑inC.
Theorem 8.2. Let C be a class of structures. Then Tn+16⊑inC implies twdn(C) < ∞.
Proof Suppose that Tn+16⊑inC. Then P 6⊑inC since Tn+1 ⊑inP. By Lemmas 6.10 and 6.11,
this implies that P * Min(Gf(C)). Therefore, we can find a path that is not in Min(Gf(C)). By Theorem 4.9, it follows that there are numbers k, l ∈N such that twdl(C) < k.
By induction on l, we prove that twdl(C) < ∞ implies twdn(C) < ∞. For l ≤ n, there
is nothing to do. For l > n, we have Tn+1⊑inTl, which implies that Tl 6⊑inC. Consequently,
it follows by Lemma 8.1 and Corollary 4.13 that twdl−1(C) < ∞. By induction hypothesis,
the result follows.
By Lemma 5.2 we obtain the following results. Corollary 8.3. If Tn+16⊑inC then C ⊑inTn.
Corollary 8.4. Tn⊳inTn+1.
To conclude the proof of Theorem 6.4 it remains to show that there are no classes between the lower part of the hierarchy and its upper part.
Lemma 8.5. Let C be a class of structures. If Tn⊑inC, for all n ∈N, then P ⊑inC.
Proof We show the contraposition. Suppose that P 6⊑in C. We have to show that there is
some n such that Tn6⊑inC. As in the proof of Theorem 8.2 it follows that there are numbers
k, l ∈ N such that twdl(C) < k. Hence, we can use Lemma 5.2 to obtain a transduction τ
witnessing that C ⊑inTl. By Theorem 7.9 we have Tl+1 6⊑inTl. It follows that Tl+1 6⊑inC, as
desired.
Corollary 8.6. If C ⊏inP then there is some n ∈N such that C ⊑inTn.
Proof By Lemma 8.5, there is some n ∈N such that Tn+16⊑inC. Hence, Corollary 8.3 implies
that C ⊑inTn.
Together, Corollaries 7.4, 8.4, and 8.6 (and the fact that every class C satisfies ∅ ⊑in
C ⊑in G) show that every class of Σ-structures is ≡in-equivalent to some of the classes in
9. Prospects and conclusion
Above we have obtained a complete description of the transduction hierarchy for classes of finite incidence structures. The most surprising result is that the hierarchy is linear. At this point there are at least three natural directions in which to proceed.
(i) We can study the hierarchy for classes of structures, instead of classes of incidence structures.
(ii) We can consider the hierarchy for classes of infinite structures. (iii) We can replace MSOby a different logic.
An answer to (ii) seems within reach, at least if we restrict our attention to countable structures. Although the resulting hierarchy is no longer linear we can adapt most of our techniques to this setting. (For an example of nonlinearity, note that the class of all countable trees and the class of all finite grids are incomparable.)
Concerning question (iii), let us remark that all results above go through if we use
CMSO instead ofMSO. We only need the right definition of rank for CMSO. In the proof of Theorem 7.9 we needed the fact that there are only finitely many theories of bounded rank. We can ensure this forCMSO by defining the rank as the least number n such that
• the nesting depth of quantifiers is at most n and
• in every cardinality predicate |X| ≡ k (mod m) we have m ≤ n.
One can check that, with this definition of rank, the proof of Theorem 3.13 also goes through for CMSO.
For logics much weaker than MSO, on the other hand, it seems unrealistic to hope for a complete description of the corresponding transduction hierarchy. For instance, a related hierarchy for first-order logic was investigated by Mycielski, Pudlák, and Stern in [MPS90]. The results they obtain indicate that the structure of the resulting hierarchy is very compli-cated.
Finally, let us address question (i). When using transductions between structures instead of their incidence structures, we can transfer some of the above results to the correspond-ing hierarchy. But we presently have no complete description since we miss some of the corresponding excluded minor results.
Lemma 9.1. Let C, S ⊆STR and suppose that S is k-sparse. (a) C ⊑inS implies C ⊑ S.
(b) S ⊑ C implies S ⊑inC.
Proof There is a transduction ̺ such that C = ̺(Cin). Since S is k-sparse we can also find a
transduction σ such that Sin= σ(S). Consequently,
Cin⊆ τ (Sin) implies C ⊆ (̺ ◦ τ ◦ σ)(S) ,
and S ⊆ τ (C) implies Sin⊆ (σ ◦ τ ◦ ̺)(Cin) .
Theorem 9.2. We have the following hierarchy:
∅ ⊏ T0 ⊏T1 ⊏· · · ⊏ Tn· · · ⊏ P ⊏ Tω⊏G ≡STR[Σ]
Proof Note that all classes in Theorem 9.2 are 2-sparse. For 2-sparse classes C and K, Lemma 9.1 implies that
C ⊑ K iff C ⊑inK .
